= Twin-prime upper bound using Bombieri–Vinogradov
Apply the <Selberg upper-bound sieve> to the numbers $p+2$ with $p\leq x$ <prime>. For <odd> <squarefree integers> $d$, the local count is $\pi(x;d,-2)$ and the density is $1/\phi(d)$. With sieve level $D=x^{1/3}$, the normalizing sum is $\gg\log x$ by the <truncated Euler-product lower bound>. The <Selberg least-common-multiple weights> have <absolute values> at most $3^{\omega(d)}$. The <weighted arithmetic-progression error bound> and <Bombieri–Vinogradov theorem> make their total error $O(x/\log^3x)$. Consequently the number of <twin prime> pairs with smaller member at most $x$ is $O(x/\log^2x)$.
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